The *Dirichlet divisor problem* has a history of such minor improvements, each with progressively longer proof. The problem asks for possible exponents $\theta$ for which we have $\sum_{n\leq x}d(n)=x\log x+(2\gamma-1)x+O(x^\theta)$, where $d$ is the divisor-counting function. It is known $\theta<\frac{1}{4}$ can't work, and it's conjectured any $\theta>\frac{1}{4}$ does.

Progress towards showing this has been rather slow: Dirichlet has shown $\theta=\frac{1}{2}$ works, Voronoi has improved it to $\theta>\frac{1}{3}$ and since then we had around a dozen of papers, each more difficult than the previous one, none of which has improved $\theta$ by more than $0.004$, see [here](https://en.wikipedia.org/wiki/Divisor_summatory_function#Dirichlet's_divisor_problem) for details.

Similar story happens with Gauss circle problem, see the table [here](http://mathworld.wolfram.com/GausssCircleProblem.html).